Nikol’skii’s inequality for different metrics and properties of the sequence of norms of the Fourier sums of a function in the Lorentz space

Let (X, Y) be a pair of normed spaces such that X ⊂ Y ⊂ L1[0, 1]n and {ek}k be an expanding sequence of finite sets in ℤn with respect to a scalar or vector parameter k, k ∈ ℕ or k ∈ ℕn. The properties of the sequence of norms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\{ \left\| {S_{e_k } (f)} \right\|x\} _k $$ \end{document} of the Fourier sums of a fixed function f ∈ Y are studied. As the spaces X and Y, the Lebesgue spaces Lp[0, 1], the Lorentz spaces Lp,q[0, 1], Lp,q[0, 1]n, and the anisotropic Lorentz spaces Lp,q*[0, 1]n are considered. In the one-dimensional case, the sequence {ek}k consists of segments, and in the multidimensional case, it is a sequence of hyperbolic crosses or parallelepipeds in ℤn. For trigonometric polynomials with the spectrum given by step hyperbolic crosses and parallelepipeds, various types of inequalities for different metrics in the Lorentz spaces Lp,q[0, 1]n and Lp,q*[0, 1]n are obtained.